document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=948400>Question 948400</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How would I find the x and y-intercept and vertex of: H(x)=3x-1/2x^2\r<BR>\n" );
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document.write( "Thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #578886 by </B><A HREF=/tutors/aboutme.mpl?userid=macston><B>macston(5194)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=macston><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=macston><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=578886\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  H(x)=3x-1/2x^2 Write the equation in the form:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In this case:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%2F2%29x%5E2%2B3x=y ALIGN=MIDDLE  ALT=\"%28-1%2F2%29x%5E2%2B3x=y\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " The x-intercept is where y=0 so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%2F2%29x%5E2%2B3x%2B0=0 ALIGN=MIDDLE  ALT=\"%28-1%2F2%29x%5E2%2B3x%2B0=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.5x%5E2%2B3x%2B0+=+0 ALIGN=MIDDLE  ALT=\"-0.5x%5E2%2B3x%2B0+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%283%29%5E2-4%2A-0.5%2A0=9 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%283%29%5E2-4%2A-0.5%2A0=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=9 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28-3%2B-sqrt%28+9+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-3%2B-sqrt%28+9+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%283%29%2Bsqrt%28+9+%29%29%2F2%5C-0.5+=+0 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%283%29%2Bsqrt%28+9+%29%29%2F2%5C-0.5+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%283%29-sqrt%28+9+%29%29%2F2%5C-0.5+=+6 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%283%29-sqrt%28+9+%29%29%2F2%5C-0.5+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.5x%5E2%2B3x%2B0 ALIGN=MIDDLE  ALT=\"-0.5x%5E2%2B3x%2B0\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.5x%5E2%2B3x%2B0+=+-0.5%28x-0%29%2A%28x-6%29 ALIGN=MIDDLE  ALT=\"-0.5x%5E2%2B3x%2B0+=+-0.5%28x-0%29%2A%28x-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 0, 6.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-0.5%2Ax%5E2%2B3%2Ax%2B0+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-0.5%2Ax%5E2%2B3%2Ax%2B0+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "The answers are 0 and 6, so the x-intercepts are (0,0) and (6,0)<BR>\n" );
document.write( "And the y-intercept(s) is where x=0, so:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F2%290%5E2%2B3%280%29 ALIGN=MIDDLE  ALT=\"y=%28-1%2F2%290%5E2%2B3%280%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "y=0 So the y-intercept is at the origin (0,0) <BR>\n" );
document.write( "The axis of symmetry is given by: (This is the x value of the vertex)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-b%2F2a ALIGN=MIDDLE  ALT=\"x=-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In this case:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-3%29%2F%282%28-1%2F2%29%29 ALIGN=MIDDLE  ALT=\"x=%28-3%29%2F%282%28-1%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x=3 Substitute this in the original equation to get the y value of the vertex.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F2%293%5E2%2B3%283%29 ALIGN=MIDDLE  ALT=\"y=%28-1%2F2%293%5E2%2B3%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F2%29%289%29%2B9 ALIGN=MIDDLE  ALT=\"y=%28-1%2F2%29%289%29%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-4.5%2B9 ALIGN=MIDDLE  ALT=\"y=-4.5%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "y=4.5 So the vertex is (3,4.5)\r<BR>\n" );
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document.write( "  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );